Goodman-Xin [1]} first verified the viscous limit for
piecewise smooth solutions separated by non-interacting shock waves
using a matched asymptotic expansion method. Later Yu [2]}
proved it for the corresponding hyperbolic conservation laws with
both shock and initial layers. In 2005, important progress made by
Bianchini-Bressan[3]} justifies the vanishing viscosity limit
in BV space even though the problem is still unsolved for the
physical system such as the compressible Navier-Stokes equations.
For the compressible isentropic Navier-Stokes equations (REF ),
Hoff-Liu [4]} first proved the vanishing viscosity limit
for piecewise constant shock even with initial layer. Later Xin
[5]} obtained the zero dissipation limit for rarefaction
waves without vacuum for both rarefaction wave data and
well-prepared smooth data. Then Wang [6]} generalized the
result of Goodmann-Xin [1]} to the isentropic
Navier-Stokes equations (REF ). For the full Navier-Stokes
equations where the conservation of the energy is also involved,
there are also many results on the zero dissipation limit to the
corresponding full Euler system with basic wave patterns without
vacuum. We refer to Jiang-Ni-Sun [8]} and Xin-Zeng
[9]} for the rarefaction wave, Wang [10]} for the
shock wave, Ma [11]} for the contact discontinuity and
Huang-Wang-Yang [12]}, [13]} for the
superposition of two rarefaction waves and a contact discontinuity
and the superposition of one shock and one rarefaction wave cases.
